Question

Find a formula for the exponential function passing through the points

4
3,
and (1,20)
125
f(x) -

Answer 1

`y = 4*5^x`

Explanation:

Given

`(x_1,y_1) = (-3,(4)/(125))`

`(x_2,y_2) = (1,20)`

Required

Determine the exponential equation

An exponential equation is of the form:
`y = ab^x`

In:
`(x_2,y_2) = (1,20)`

`20 = ab^1`

`20 = ab` ---- (1)

In:
`(x_1,y_1) = (-3,(4)/(125))`

`(4)/(125) = ab^(-3)` --- (2)

Divide (1) by (2)

`20/(4)/(125) = (ab)/(ab^(-3))`

`20/(4)/(125) = b^4`

`20*(125)/(4) = b^4`

`5*125 = b^4`

`625 = b^4`

Take 4th roots of both sides

`\sqrt[4]{625} = b`

`5 = b`

`b = 5`

Substitute
`b = 5` in
`20 = ab`

`20 = a * 5`

Solve for a

`a = 20/5`

`a = 4`

Hence, the equation is:

`y = 4*5^x`